345
21.3. CP violation in neutral K-meson decays
freedom which allow this analysis to determine the penguin contributions and
the strong phases, and hence α. However, the resonance overlap regions cover
a small fraction of the Dalitz plot, so that a substantial data sample (a few
thousand events) is needed to constrain all the amplitude parameters.
An isospin analysis similar to that of the ππ states can be done for the
ρπ states, but now there is no reason to forbid the final state to have I = 1.
Nevertheless, if charged B decays are also included, there are five physical
amplitudes (ρ
0
→ π
+ π
− , π
− π
+ , π
0 π
0 , ρ
+
→ π
+ π
0 , ρ
−
→ π
− π
0 ) which are
expressible in terms of two pure tree (ΔI = 3/2) transitions to I = 1, 2 final
states. One of the pure tree amplitudes may be written (Gronau 1991) as the
sum A
+ + A
− + 2A
0 , and hence the ratio ( A ¯ + + A ¯ − + 2A ¯ 0 )/(A
+ + A
− + 2A
0 )
has the phase 2α.
This approach has been followed by both BaBar and Belle, with the results
BaBar (Aubert et al. 2007b)
α = 87
+45
◦
(21.69)
−13 ◦
Belle (Kusaka et al. 2007)
68 < α < 95
◦ .
(21.70)
These results are consistent with the values of β and γ given in (21.52), (21.25)
and (21.26), given the definition α = π − β − γ.
Of course, this is only one (at present not very tight) consistency check.
But there are now very many independent measurements of the magnitudes of
the CKM matrix elements, as well as the angles. We shall not describe these
here, referring the reader to the regular updates by the Particle Data Group
(currently Nakamura et al. 2010). We showed in figure 20.11 the 2010 plot of
the contraints in the ¯
ρ, η ¯ plane, presented by Ceccucci et al.. They concluded
that the 95% CL regions all overlapped consistently around the global fit region, though the consistency of |V
ub
| and sin 2β was not very good. It would
be premature to make too much of the minor reservation, though it may be
noted that sin 2β could be sensitive to new physics via short-distance corrections to the box diagrams of figure 21.5, while |V ub | is obtained from a treelevel process, and is thus unlikely to be affected by new physics. Overall, the
consistency represented in figure 20.11 must be counted as a major triumph
of the Standard Model, in particular of the original analysis by Kobayashi
and Maskawa (1973). It must be remembered, though, that many extensions
of the Standard Model allow considerable room for new CP-violating effects,
which could be revealed by increasingly precise determinations of the CKM
parameters.
21.3 CP violation in neutral K-meson decays
Although the formalism is similar, the phenomenology of CP violation in neutral K-meson decays is very different from that in neutral B-meson decays.
In the K case, CP violation is a very small effect, typically at the level of
21.3. CP violation in neutral K-meson decays
freedom which allow this analysis to determine the penguin contributions and
the strong phases, and hence α. However, the resonance overlap regions cover
a small fraction of the Dalitz plot, so that a substantial data sample (a few
thousand events) is needed to constrain all the amplitude parameters.
An isospin analysis similar to that of the ππ states can be done for the
ρπ states, but now there is no reason to forbid the final state to have I = 1.
Nevertheless, if charged B decays are also included, there are five physical
amplitudes (ρ
0
→ π
+ π
− , π
− π
+ , π
0 π
0 , ρ
+
→ π
+ π
0 , ρ
−
→ π
− π
0 ) which are
expressible in terms of two pure tree (ΔI = 3/2) transitions to I = 1, 2 final
states. One of the pure tree amplitudes may be written (Gronau 1991) as the
sum A
+ + A
− + 2A
0 , and hence the ratio ( A ¯ + + A ¯ − + 2A ¯ 0 )/(A
+ + A
− + 2A
0 )
has the phase 2α.
This approach has been followed by both BaBar and Belle, with the results
BaBar (Aubert et al. 2007b)
α = 87
+45
◦
(21.69)
−13 ◦
Belle (Kusaka et al. 2007)
68 < α < 95
◦ .
(21.70)
These results are consistent with the values of β and γ given in (21.52), (21.25)
and (21.26), given the definition α = π − β − γ.
Of course, this is only one (at present not very tight) consistency check.
But there are now very many independent measurements of the magnitudes of
the CKM matrix elements, as well as the angles. We shall not describe these
here, referring the reader to the regular updates by the Particle Data Group
(currently Nakamura et al. 2010). We showed in figure 20.11 the 2010 plot of
the contraints in the ¯
ρ, η ¯ plane, presented by Ceccucci et al.. They concluded
that the 95% CL regions all overlapped consistently around the global fit region, though the consistency of |V
ub
| and sin 2β was not very good. It would
be premature to make too much of the minor reservation, though it may be
noted that sin 2β could be sensitive to new physics via short-distance corrections to the box diagrams of figure 21.5, while |V ub | is obtained from a treelevel process, and is thus unlikely to be affected by new physics. Overall, the
consistency represented in figure 20.11 must be counted as a major triumph
of the Standard Model, in particular of the original analysis by Kobayashi
and Maskawa (1973). It must be remembered, though, that many extensions
of the Standard Model allow considerable room for new CP-violating effects,
which could be revealed by increasingly precise determinations of the CKM
parameters.
21.3 CP violation in neutral K-meson decays
Although the formalism is similar, the phenomenology of CP violation in neutral K-meson decays is very different from that in neutral B-meson decays.
In the K case, CP violation is a very small effect, typically at the level of